1. Introduction
As one of the most important structural engineering materials, concrete has been widely used in various engineering fields. Among them, the cracking of mass concrete structures caused by temperature is also being paid more and more attention to by people in the fields of water conservancy and civil engineering. At present, the research on the working behavior and cracking of concrete structures under mechanical load is relatively thorough, and the research on the cracking of concrete structures at the macroscopic scale caused by temperature has also made many achievements, see, e.g., [
1,
2,
3,
4,
5,
6,
7,
8]. However, concrete is a multiphase composite material consisting of cement, sand, stone, and water. Thus, it may have some limitations in analyzing the deformation and cracking failure processes of concrete under load by ignoring its heterogeneity and assuming it to be a homogeneous material at the macroscopic level.
The mesoscopic mechanics method regards concrete as a three-phase heterogeneous composite material composed of aggregate, mortar matrix, and ITZ between the aggregate and the mortar matrix. By adopting an appropriate concrete mesoscopic structural model and dividing the finite element meshes at the mesoscopic level, the crack propagation and failure pattern of the concrete specimen can be simulated intuitively. Thus, the mesoscopic mechanics method is regarded as an effective way to solve the problem of concrete failure. It has been the subject of unprecedented attention and concern by many researchers and has achieved fruitful results. Buttignol et al. [
9] established a concrete mesoscopic model by the Monte Carlo method to study the instantaneous creep problem of the compressive concrete specimen under high temperature conditions. The results show that the numerical simulation results are in good agreement with the test results, and the boundary conditions have an important effect on the total deformation of the concrete specimen. Wu et al. [
10] employed the take-and-place approach to establish a reliable mesostructure model of concrete. In their work, based on the mesoscale model of concrete, not only the mechanical problems but also the effects of the interface debonding on the thermal conduction as well as humidity diffusion were investigated. By using the Monte Carlo method, Zhou et al. [
11] established a random polygon mesoscale model of concrete, within which the mortar, ITZ, and aggregates are distinctly represented. The results show that the permeability of the ITZ and the aggregate volume fraction play a significant role in the freezing and thawing behavior of concrete. Chen et al. [
12] proposed a heuristic packing algorithm for constructing 3D random aggregate models with high aggregate content. The results show that the algorithm is more efficient and robust. By using an in-house code based on Voronoi tessellation and splining techniques, Naderi et al. [
13] developed a 3D mesoscale model of concrete composed of real-shape coarse aggregates, mortar, and ITZ. The results show that the irregular shape of the aggregate plays an important role in the micro-crack nucleation and ultimate fracture pattern. Rodrigues et al. [
14] proposed a new approach for concurrent multiscale modeling of 3D crack propagation in concrete. By using this method, a macroscopic model with homogenized elastic parameters was adopted in the regions where the material behaves elastically. For regions where cracks were expected to occur, a mesoscopic model based on a mesh fragmentation technique was used to represent the concrete as a heterogeneous three-phase material composed of mortar matrix, aggregate, and ITZ. The simulation results show that the method can not only greatly reduce the calculation scale of the model, but also the propagation path of the crack is more consistent with the crack propagation path of the real concrete specimen. Thilakarathna et al. [
15] summarized and commented on the evolution and research status of the concrete random aggregate model. They suggested scanning concrete specimens with laser scanning or computed tomography (XCT) technology to establish a real concrete particle structure and then performing structural calculations and analysis, which is a hot topic in current research. However, because of the complexity of generating a concrete random aggregate model and the high computational cost, only small-sized concrete specimens can be modeled, and large concrete structures cannot be modeled. In addition, many other researchers, such as the literature [
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30], have undertaken extensive studies on concrete mesoscopic mechanical models and developed various aggregate generation methods and placement algorithms.
Although the aggregate generation method and placement algorithm proposed by these studies can make the aggregate shape very close to the real concrete aggregate form and the aggregate filling amount can also reach a high degree, most of these studies need to program the aggregate generation and placement algorithm in Matlab, Fortran, Python, or C language to accomplish the aggregate placement. To complete the model visualization and finite element computation, researchers should employ additional finite element software, which requires a high level of programming and the development of a more complicated interface program to dock with this finite element software. Docking between programs usually requires transferring a large number of parameters, which occupies more computer memory. As a result, the demands on computer performance are considerable. Moreover, most of these studies are focused on cracking simulation of small-sized concrete specimens due to cost and computation time constraints. However, relevant reports for the simulation of crack propagation of 3D mass concrete structures with a large number of finite element meshes are relatively rare.
The finite element software ANSYS provides powerful pre-and post-processing capabilities. It also has an APDL programming language, similar to Fortran 77, with over 1000 commands. The APDL language can not only do most of the operations of the Fortran 77 language but can also perform other operations such as element selection, node selection, local coordinate system building, and rotation, which are not needed for secondary development by using other programming languages [
31]. Therefore, researchers only need to master the APDL language to complete the programming implementation of aggregate generation and placement algorithms, as well as model visualization and finite element calculations, without having to learn additional languages or develop complex interface programs. Because of this advantage of the APDL language, many researchers have turned to this to do research on mesoscopic concrete models (e.g., [
32,
33,
34]). Using the APDL language of ANSYS software, there are currently two ways to create a concrete mesoscopic mechanical model: first, the aggregate and ITZ boundaries are generated using Boolean operations, which are separated from the mortar boundary, and then the aggregate, ITZ, and mortar are meshed separately; second, the “background grid” method is used to generate aggregate and ITZ boundaries. That is, the mortar matrix is divided into finite element meshes first, and then the material properties of each element are judged by calculating the distance between all nodes on the element from the center of the aggregate sphere, and the corresponding material number is assigned to the element, so as to complete the aggregate placement. Although using these two methods to create a mesoscopic model of concrete has many of the benefits described above, it is worth noting that when using the first method (i.e., Boolean operations) to generate aggregates, mesh distortion is generally not caused if the meshing is carried out for large particle size particles, but when the aggregate size is small, it will often cause mesh distortion, making the calculation non-convergent and the program to report errors; when the second method is used, it is necessary to create a 3D array to store the node coordinate information of the element, and then judge the material properties of the element by calculating the distance from the node to the center of the aggregate sphere. This method can generate an aggregate model that meets the requirements if the number of elements is small, but if the number of elements reaches millions, the storage operation of the 3D array will occupy a large amount of computer memory space, reducing computing efficiency and increasing computer time, which is a significant challenge to the performance of the computer. Obviously, the random aggregate model of concrete established by the above two methods is also difficult to promote in practical engineering applications, especially in the damage-fracture analysis of high concrete dams. There are currently no relevant literature reports available.
From the above analysis, it can be seen that the existing research on mesoscopic mechanics method of concrete is usually limited to the simulation calculation of small-scale concrete specimens, while the cracking simulation of large-scale concrete structures, such as concrete gravity dams, is much less. To address the aforementioned challenges, this work presents a new approach for completing random aggregate placement using the ESEL command in the APDL language and adjusting the rotation angle of the local coordinate system.
When the method proposed in this paper is used, the placement boundary of aggregate is first determined, and then the radius information and the center coordinate information of aggregate are randomly generated by using the random number generation function RAND in the APDL language, and stored in a 2D array. Then, the mesh of the model is divided. After the mesh division is completed, the aggregate center coordinates and their radius information stored in the array are read, and the local coordinate system definition command LOCAL in the APDL language is used to define the local cartesian coordinate system, spherical coordinate system, or ellipsoidal coordinate system at any aggregate center coordinates. After the local coordinate system has been built, the ESEL command in the APDL language is used to screen the element set that meets the requirements of aggregate radius, and the material attribute of aggregate is given to this element set, thus completing the placement of an aggregate. By using the loop statement, traverse all aggregates stored in the array and read the corresponding aggregate center coordinates and radius data. According to the method mentioned above, all aggregates can be generated. As the APDL language of ANSYS software is used to complete the aggregate placement by using the method in this paper, and the ANSYS software platform is also used to complete the subsequent finite element calculation, the researchers can complete the aggregate generation, placement, model visualization, and finite element calculation without using other languages and compiling complex interface programs. Moreover, because there is no need to transfer a large number of parameters between different programs, the occupation of computer memory space can be greatly reduced. In addition, unlike some existing methods described above, when using the method in this paper, since the material properties of each element are not judged by calculating the distance from the node of each element to the center of the aggregate sphere, there is no need to define an oversized 3D array to store the node coordinate information of the element. As mentioned above, when the number of element nodes is large, the storage operation of the 3D array will occupy a large memory space on the computer, which is a great challenge to the performance of the computer, and this difficulty can be overcome by adopting the method proposed in this paper. To sum up, compared with the previous concrete aggregate placement methods, the use of this method in the process of concrete aggregate placement and subsequent finite element calculation can greatly reduce the occupation of computer memory space, improve the calculation efficiency, and make it possible to apply the mesoscopic mechanics method to the finite element calculation of large-scale 3D concrete structures.
In terms of material models, the material damage model or fracture model of each phase (e.g., [
9,
10,
13,
29]) that is used in the majority of current research on damage and fracture simulation of mesoscopic models of concrete is based on a macroscopic phenomenological method. That is, rather than the physical background of damage and fracture, or the mesoscopic structural changes within the material, it focuses on the effect of damage and fracture on the macroscopic mechanical properties of the material. Since the macroscopic phenomenological method simulates the macroscopic mechanical behavior of materials from macroscopic phenomena, the determination of the equations and parameters of these material models is usually half empirical and half theoretical, and has clear physical significance, which can directly reflect the stress state of structures. However, as described in the literature [
35], the damage and failure of concrete are caused by the growth and accumulation of micro-cracks and micro-pores of various scales within it. Especially when considering temperature factors, due to the difference in the thermal expansion coefficient between aggregate and mortar, thermal incompatibility stress will be generated at the ITZ between mortar and aggregate, which will result in micro-cracks at the ITZ. When micro-cracks at the ITZ accumulate to a certain degree, it may extend to mortar and aggregate. Therefore, from the perspective of meso-mechanics, the micro-cracks, micro-defects, and other damage of concrete materials are interwoven and influenced by each other. When using the meso-mechanics method to investigate concrete damage and fracture problems, there are some limitations in rationality if the damage or fracture model used as material constitutive model is still derived using the traditional macroscopic phenomenological method, especially when considering temperature load. Therefore, an isotropic temperature damage model that takes into account the relationship between various phases of concrete has been proposed in the literature [
35]. Considering the characteristics of anisotropic damage of concrete, this paper refers to relevant references and makes some improvements to this isotropic temperature damage model through a series of derivations, so an improved anisotropic temperature damage model is presented.
Based on the random aggregate placement method of concrete proposed in this paper, a 3D macro–meso finite element model of Shimantan concrete gravity dam is established in this paper, and the damage of the dam under two conditions of sudden temperature drop and sudden temperature rise is numerically simulated with the anisotropic temperature damage model presented in this paper, which takes into account the relationship between various phase materials of concrete. In order to verify the effectiveness of the aggregate placement method proposed in this paper, in
Section 3, some small-scale concrete specimens are used to verify the placement efficiency and aggregate morphology. In addition, in order to verify the effectiveness of the anisotropic temperature damage model, the macro model and the bilinear damage model obtained based on the macro phenomenological method are also used to numerically simulate the temperature damage of Shimantan dam under the two working conditions. The results show that the method in this paper is feasible and can provide some references for other researchers to study similar projects.
4. Material Constitutive Model
Based on the strain equivalence hypothesis [
36], the damage constitutive model under uniaxial conditions can be expressed as follows:
where
,
,
,
and
denote the nominal stress, the effective stress, the damage variable, the initial Young’s modulus and strain, respectively.
According to Equation (6), many damage evolution equations, including isotropic and anisotropic models, have been proposed by scholars from various countries, such as references [
37,
38,
39,
40,
41]. However, as stated in the introduction section, these damage models are obtained on the basis of assuming that concrete is a homogeneous material and using the traditional macroscopic phenomenological method without considering the mutual influence of the damage between each phase of concrete. When using the mesoscopic mechanics method, which regards concrete as a multi-phase material consisting of aggregate, mortar, and ITZ, to simulate the damage and fracture of concrete, it is obviously inconsistent with the actual situation when using the damage model obtained from the macroscopic phenomenological method to simulate damage. Therefore, the literature [
35] proposed a temperature damage model considering the relationship between each phase of concrete. The specific expression is:
where
and
are the damage variables of mortar and ITZ in concrete, respectively. The value range of the damage variables is
, if
, the material is not damaged, and if
, it indicates that the material is completely fractured.
,
and
represent the fracture energy, elastic modulus, and tensile strength of the mortar, respectively.
,
and
represent the fracture energy, elastic modulus, and tensile strength of the ITZ, respectively.
stands for the change of temperature.
and
can be expressed as follows:
where
and
are the fracture constants. Referring to the literature [
42,
43], in this paper, the values of the two constants are set to 1.414 and 0.2324, respectively.
,
and
represent the thermal expansion coefficient, elastic modulus, and Poisson’s ratio of the mortar, respectively.
,
and
denote the thermal expansion coefficient, elastic modulus, and Poisson’s ratio of the aggregate, respectively.
As stated in the literature [
38,
39], the damage of concrete material has obvious anisotropic characteristics. The damage evolution equation represented by Equation (7) reflects the damage condition of materials in one-dimensional (1D) space and is isotropic. In order to reflect the damage anisotropy of concrete material, Equation (7) needs to be extended to 3D space and be directional. When the anisotropic damage variable is used instead of the scalar damage variable, Equation (6) needs to be extended accordingly and the corresponding stiffness degradation matrix should be defined. After being extended, Equation (6) can be expressed as follows:
where
is the effective stress tensor,
is the nominal stress tensor,
is the second-order identity tensor, and
is the second-order damage tensor, which can be expressed as follows:
where
is the main direction of damage vector and
is the principal value of the damage tensor
. In the principal damage space, the matrix representation of the tensor
can be expressed as follows:
where
represents the eigenvalues and can be expressed as:
with
being an unknown quantity to be determined by the strain-based damage evolution function. In the subsequent development, the superscript hat symbol
denotes a principal value of
.
As described by Murakami [
44], the effective stress tensor obtained by the strain equivalence hypothesis (i.e., Equation (9)) is asymmetric. Since it is usually inconvenient to use the asymmetric effective stress tensor in the formulation of constitutive and evolution equations, several symmetrized methods have been proposed by many authors. Among these methods, one can use the strain energy equivalence hypothesis to overcome this limitation. However, as stated in the literature [
29], using the strain energy equivalence hypothesis will complicate the constitutive model. The multiplicative decomposition symmetrization method proposed by the literature [
45] is used frequently, such as in the literature [
39,
46,
47,
48]. By this method, Equation (9) can be rewritten as follows:
or
is a fourth-order damage-effect tensor and can be expressed as:
In the principal coordinate system of damage
with the Voigt notations, the fourth-order damage-effect tensor
can be expressed as the following “diagonal matrix form”:
where
with
being the principal damage variable.
Since the fourth-order damage-effect tensor
contained more individual components, some authors [
49,
50] assumed that the principal axes of damage coincide with the principal axes of stress and strain and used the simplified damage effect tensor in Equation (16) to establish the relation between the effective and nominal stress tensors, and the matrix representation of the nominal stress tensor can be written as:
where
is the initial undamaged elastic stiffness matrix and
is the damaged elastic stiffness matrix which can be expressed as:
with
,
and
.
In addition,
and
in Equation (17) can be expressed as:
The damage–effect matrix shown in Equation (16) is in the principal coordinate system of damage, which corresponds to the direction of the principal strains. In terms of a general coordinate system, it should be transformed to the global coordinate system as follows:
where
By substituting Equation (20) into Equation (17), Equation (17) can be rewritten as:
Then the nominal stress calculated by Equation (22) is the physical quantity in the global coordinate system. After the damaged elastic stiffness matrix has been defined, it is necessary to determine the damage evolution equation that can calculate the anisotropic damage variables
It can be seen from Equation (7) that the temperature damage evolution equation proposed in reference [
35] is a function of temperature, which is a scalar variable and has no direction. Therefore, it cannot well express the anisotropic damage characteristics of concrete under the multiaxial stress state and should be extended to a multi-axial form. As mentioned in the literature [
51], the surface deformation of the concrete dam blocks caused by temperature change will be constrained by the concrete inside the dam blocks, and the constraint coefficient can be expressed as:
where
is the constraint coefficient,
is the thermal expansion coefficient of the concrete, and
denotes the temperature increment.
According to Equation (23), one can get that:
By substituting Equation (24) into Equation (7), Equation (7) can be rewritten as follows:
where
and
represent the thermal expansion coefficient of the mortar and ITZ, respectively.
Since the strain
in Equation (25) is a scalar, the damage variables obtained from Equation (25) are still isotropic variables. Considering the characteristics of anisotropic damage, the principal strain
can be used to replace the strain
in the Equation (25) to make the damage evolution equation have direction. Then Equation (25) can be changed to:
where
;
is the principal strain;
and
are the damage values of mortar and ITZ in the three principal strain directions, respectively;
is the constraint coefficient, which is set to 0.95 in this paper based on the literature [
51].
Obviously, the damage evolution equation expressed in Equation (26) can reflect the damage anisotropy of concrete material, but when the principal strain is compressive strain, the calculated damage value may deviate from the actual situation. As stated in the literature [
39,
52], the damage of concrete material is mainly caused by tensile strain. When the material is under tensile loading, the micro-cracks propagate perpendicular to the direction of tension. The direction of damage is the same as the direction of tension and is called “direct damage.” The damage caused by material under compression is mainly caused by lateral tension strain due to the Poisson effect. The direction of damage is orthogonal to the direction of compression, and the direction of microcrack propagation is parallel to the direction of loading, which is called “indirect damage.” Therefore, further improvement is required for Equation (26).
When the finite element software is used for structural calculation, the user-defined constitutive model can be realized via the user material subroutine of the software. ABAQUS software provides user material subroutine UMAT, ANSYS software provides user material subroutine USERMAT, and some other finite element software also provides corresponding user material subroutines, which will not be listed one by one here. When users use these material subroutines to define their own material constitutive model, the Jacobian matrix of material constitutive, which is the change rate of stress increment corresponding to strain increment, needs to be provided. For linear elastic material constitutive models, such as the linear elastic damage model defined in this paper, the Jacobian matrix is
in Equation (22). For other elastic–plastic or more complex models, the corresponding Jacobian matrix can be defined according to the characteristics of the model. After the Jacobian matrix is defined, the main program usually uses the geometric relationship to calculate the strain increment
of each incremental step or iteration, which is transmitted to the user material subroutine. The user material subroutine calculates the stress increment of each incremental step or iteration according to the Jacobian matrix and obtains the updated stress and strain with the following equation:
where
represents incremental step.
According to the above calculation principle, under triaxial stress states, the normal stress of any point in the structure along one coordinate axis will be affected by the strain in the other two coordinate axes that are orthogonal to it when the stress–strain calculation and analysis of the structure are carried out by the finite element method. The value of the normal stress along any coordinate axis is actually the result of the strain superposition along the three coordinate axes. Any principal stress at this point is also the result of the superposition of three principal strains. When damage is not considered, the principal stress at any point in the principal stress space can be calculated by the following formula:
According to the general Hooke’s law, the strain components in the three principal directions are satisfied with the following relationships:
where
is the strain in the
i-direction caused by the stress in the
i-direction;
is the strain in the
i-direction caused by the stress in the
j-direction due to the Poisson effect;
is the strain in the
i-direction caused by the stress in the
k-direction due to the Poisson effect.
According to Equation (29), Equation (28) can also be expressed as:
It can be seen from Equation (29) that when
, one can get
. This implies that the magnitude of a principal stress is only related to the principal strain under the uniaxial stress state, which is different from the triaxial stress state. Therefore, Equation (30) can actually be regarded as the expression of the stress–strain relationship of three principal stresses under the uniaxial stress state. Combining Equation (28), one can get that, under the triaxial stress state, when the three principal strains are all greater than 0 (i.e.,
), any principal stress
reaches a certain value, such as the peak strength of the material, the required value of the strain
may be smaller than the strain
under the uniaxial stress state. This is because under the triaxial stress state, the principal stress
is the result of the superposition of the principal strains in three directions, and its corresponding strain can be expressed as
. However, under the uniaxial state, the principal stress
in one direction is only related to the strain
in the corresponding direction. When the principal strain in one direction is tensile strain and the other two directions are compressive strain (i.e.,
), the required value of the strain
for the principal stress
to reach the peak strength may be larger than the strain
under the uniaxial stress state. Equation (7) proposed by reference [
35] is the uniaxial damage evolution equation derived from the test results under the uniaxial stress state. Obviously, using Equation (25) improved from Equation (7) to calculate the damage under the uniaxial stress state will be consistent with the test results, but under the multiaxial stress state, only by giving the directionality of strain in Equation (26) to calculate the damage value in all directions, it may be greatly different from the test results. For example, under the multiaxial stress state, when the three principal strains are greater than 0 (i.e.,
), and at the time that the strain in a certain direction reaches
, the stress
in the corresponding direction calculated by the program reaches the peak strength of concrete, but under the uniaxial state, the stress calculated by this strain value
may not reach the peak strength of concrete. At this time, the damage value
calculated by substituting the strain value
into Equation (26) will be smaller than the actual damage value. This is because in the actual uniaxial test, when the stress reaches the peak strength, the strain corresponding to the real damage degree inside the material is the peak strain, while the aforementioned strain
, which is substituted into Equation (26) to calculate the damage value, has not reached the peak strain, the calculated damage value cannot reflect the true damage degree of the structure. In other words, according to the stress calculation results, the tensile strength of the material has been reached in one direction, which indicates that fracture failure may have occurred in the structure, but the damage calculation results indicate that the damage of the structure does not reach the extent of fracture failure.
It can be seen from Equation (29) that for the same stress value , is the strain corresponding to the stress value under the triaxial stress state, whereas is the strain corresponding to the stress value under the uniaxial stress state. If substituting for in Equation (26), then the calculated principal damage results can match the uniaxial test results and truly reflect the damage degree of the structure. Based on the above analysis, can be defined as follows:
- 1.
When all principal stresses are positive, i.e., tension–tension–tension mode, then can be expressed as follows:
where
;
is the principal strain in
i-direction calculated by the user material subroutine;
and
are the principal stresses in
j- and
k-directions calculated by the user material subroutine, respectively.
- 2.
When two principal stresses are positive and one principal stress is negative, i.e., tension–tension–compression mode, it can be seen from Equation (30) that the strain corresponding to the direction of compressive stress must be negative. As mentioned above, the damage in the compression direction is mainly caused by the lateral tensile strain due to the Poisson effect. The damage direction is orthogonal to the compression direction, and the microcrack propagation direction is parallel to the loading direction. Because the lateral deformation caused by the tensile stress is usually compressive deformation, which makes little contribution to the damage in the direction of compressive stress. Therefore, it is assumed that the damage in the direction of compression under the tension-tension-compression state is only related to the tension strain caused by the Poisson effect in the orthogonal direction but not to the strains in the other two directions. In addition, then can be expressed as follows:
where
;
is the tensile damage strain corresponding to tensile stress;
is the equivalent transfer tensile damage strain corresponding to compressive stress; the sign
indicates Macauley bracket, which is defined as
.
- 3.
When one principal stress is positive and the other two principal stresses are negative, i.e., tension–compression–compression mode, the damage in one compressive direction is mainly caused by the combined action of lateral deformation caused by the Poisson effect in the other two directions. In addition, in this case, the other compressive direction contributes to lateral tensile deformation while the direction of tensile stress contributes to lateral contraction deformation. In this state, the following equation is used to express in this paper:
where
.
- 4.
When all three principal stresses are negative, i.e., compression–compression–compression mode, the damage in one direction is mainly caused by the combined action of lateral deformation caused by the Poisson effect in the other two directions. The other two compressive directions contribute to lateral tensile deformation in this direction. Therefore, in this state, can be expressed as follows:
where
.
By substituting
defined by Equations (31)–(34) into Equation (26), one can get the following expression:
The damage evolution equation represented by Equation (35) can reflect the damage anisotropy of concrete material. By substituting the calculation result of into Equation (22), the elastic stiffness matrix considering damage can be calculated and the stress–strain calculation of the structure can be completed. Equation (22) and Equation (35) are the main equations of the anisotropic temperature damage model for concrete proposed in this paper. Finally, the proposed model is implemented in the ANSYS software via the user subroutine USERMAT.